THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6 Spécialité ...

64 Asymptotically exact minimax estimation in sup-norm for additive models- for d = 2, N(h) ≤ D 10(1 + (log n) D 11n 12(2 ˜β+1)) 2,- for d ≥ 3, N(h) is bounded from above by a constant.This finishes the proof.Now the upper bound (4.8) is proved exactly as in Section 3.3 of Chapter 3, replacingPropositions 3.1 and 3.2 of Chapter 3 by Propositions 4.1 and 4.2.4.4 Lower boundThe proof of the inequality (4.9) is similar to the proof of the lower bound for anisotropicHölder classes (cf. Chapter 3). The only difference is the family of functions f j used inthe proof. Here in place of the functions f j , we consider functions g j defined for t ∈ R dand j ∈ {1, . . . , M} byg j (t) = ∑ i∈Λh˜βi L iwith the h i defined as in Section 4.3, M =(1 −∣∣t i − a j,i ∣∣∣˜β)h i+[ ( ) ]1n 2 ˜β+1, where [x] denotes the integerlog npart of x and a j,i = 2(2 1/˜β + 1)jh i for j ∈ {1, . . . , M} and i ∈ Λ. We define the pointsa j by a j = (a j,1 , . . . , a j,d ) for j ∈ {1, . . . , M}. The points a j are chosen such that all thefunctions g j have disjoint supports.Now, as in Section 3.4, for θ = (θ 1 , . . . , θ M ) ∈ [−1, 1] M , we denote by f(·, θ) thefunction defined for t ∈ [0, 1] d byf(t, θ) =M∑θ j g j (t).j=1The functions f(·, θ) satisfy similar properties as the functions defined in the same wayin Section 3.4. Indeed we have the following proposition.Proposition 4.3. We have for all θ = (θ 1 , . . . , θ M ) ∈ [−1, 1] M :(i) the function f(·, θ) belongs to Σ ad (β, L),(ii) f(a j , θ) = θ j C ad ψ n ,(iii) if f = f(·, θ) in the model (4.2), for j ∈ {1, . . . , M}, the statistic∫gRy j =d j (t)dY tg∫R 2 d j (t)dt ,